| import Mathlib | |
| import OAI.Geometry.CubicTorus.Profiles.NormalizedCMCCharts | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| lemma normalized_CMC_common_patch_curvature {B C : Set Torus} | |
| {N M : Cover → Cover} (hpN : ∀ m x,N (x+latticeVector m)=N x) | |
| (hpM : ∀ m x,M (x+latticeVector m)=M x) | |
| {H K : ℝ} (hBch : HasNormalizedCMCCharts B N H) (hCch : HasNormalizedCMCCharts C M K) | |
| {x : Torus} (hxB : x∈B) (hx : x∈orientedSurfaceAgreement B C N M) : H=K := by | |
| obtain ⟨a,rfl⟩ := quotientMap_surjective x | |
| have hxC := hx.self_of_nhds.1.mp hxB | |
| have hNM : N a=M a := by | |
| simpa only [periodic_representative hpN,periodic_representative hpM] using hx.self_of_nhds.2 hxB | |
| obtain ⟨hn,f,hf,hf0,_,hfm,_⟩ := hBch a hxB | |
| obtain ⟨_,g,hg,hg0,_,hgm,hgc⟩ := hCch a hxC | |
| rw [←hNM] at hgm hgc | |
| have hF0 : normalGraph (N a) f 0=0 := by | |
| change (normalPlaneLinearIsometry (N a)) 0+f 0 • N a=0 | |
| rw [map_zero,hf0,zero_smul,add_zero] | |
| have hFt : Tendsto (normalGraph (N a) f) (𝓝 (0 : Plane)) (𝓝 (0 : Cover)) := by | |
| simpa only [hF0] using ((normalGraph_contDiff hf (n := N a)).continuous.continuousAt (x := 0)).tendsto | |
| have hq : Tendsto (fun y : Plane ↦ quotientMap (a+normalGraph (N a) f y)) | |
| (𝓝 0) (𝓝 (quotientMap a)) := by | |
| have hc := (continuous_quotientMap.comp ( (continuous_const (y := a)).add (normalGraph_contDiff hf (n := N a)).continuous)).continuousAt (x := 0) | |
| simpa only [Function.comp_def,Pi.add_apply,hF0,add_zero] using hc.tendsto | |
| have hfg : f=ᶠ[𝓝 (0 : Plane)]g := by | |
| filter_upwards [hfm,hq.eventually hx,hFt.eventually hgc] with y hfy hxy hgy | |
| rw [normalPlaneCoordinates_graph hn] at hgy | |
| exact normalGraph_height_eq hn ((hgy.mp (hxy.1.mp hfy.2.1)).symm) | |
| have hc := planeCMC_germ hfg | |
| rw [hfm.self_of_nhds.2.2,hgm.self_of_nhds.2.2] at hc | |
| exact neg_injective hc | |
| theorem CMC_surfaces_identity {B C : Set Torus} (hBc : IsClosed B) (hCc : IsClosed C) | |
| (hBconn : IsPreconnected B) (hCconn : IsPreconnected C) | |
| {N M : Cover → Cover} (hN : Continuous N) (hM : Continuous M) | |
| (hpN : ∀ m x,N (x+latticeVector m)=N x) | |
| (hpM : ∀ m x,M (x+latticeVector m)=M x) | |
| {H K : ℝ} (hBch : HasNormalizedCMCCharts B N H) (hCch : HasNormalizedCMCCharts C M K) | |
| (hpatch : (B∩orientedSurfaceAgreement B C N M).Nonempty) : | |
| B=C ∧ H=K ∧ ∀ x∈B,N (representative x)=M (representative x) := by | |
| obtain ⟨x,hxB,hxag⟩ := hpatch | |
| have hHK := normalized_CMC_common_patch_curvature hpN hpM hBch hCch hxB hxag | |
| subst K | |
| have hAB : B⊆orientedSurfaceAgreement B C N M := | |
| preconnected_subset_of_relative_germ_closed hBconn (isOpen_orientedSurfaceAgreement _ _ _ _) | |
| ⟨x,hxB,hxag⟩ (fun _ hy ↦ orientedSurfaceAgreement_closed_at hCc hN hM hpN hpM hBch hCch hy.2 hy.1) | |
| have hxC := hxag.self_of_nhds.1.mp hxB | |
| have hAC : C⊆orientedSurfaceAgreement C B M N := | |
| preconnected_subset_of_relative_germ_closed hCconn (isOpen_orientedSurfaceAgreement _ _ _ _) | |
| ⟨x,hxC,orientedSurfaceAgreement_symm hxag⟩ | |
| (fun _ hy ↦ orientedSurfaceAgreement_closed_at hBc hM hN hpM hpN hCch hBch hy.2 hy.1) | |
| refine ⟨Subset.antisymm (fun y hy ↦ (hAB hy).self_of_nhds.1.mp hy) | |
| (fun y hy ↦ (hAC hy).self_of_nhds.1.mp hy),rfl,?_⟩ | |
| exact fun y hy ↦ (hAB hy).self_of_nhds.2 hy | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| lemma boundaryZeroGerm_homeomorph (e : Torus ≃ₜ Torus) (S : Set Torus) (F : Torus → ℝ) : | |
| boundaryZeroGerm (e ⁻¹' S) (F∘e)=e ⁻¹' boundaryZeroGerm S F := by | |
| ext x | |
| change (∀ᶠ y in 𝓝 x,e y∈S → F (e y)=0) ↔ | |
| (∀ᶠ y in 𝓝 (e x),y∈S → F y=0) | |
| rw [←e.map_nhds_eq x] | |
| rfl | |
| lemma boundaryZeroGerm_translation (a : Cover) (S : Set Torus) (f : smoothPeriodicScalars) : | |
| boundaryZeroGerm ((fun b ↦ b+quotientMap a) ⁻¹' S) | |
| (fun b ↦ (sibling_smoothScalarTranslate a f).val (representative b))= | |
| (fun b ↦ b+quotientMap a) ⁻¹' boundaryZeroGerm S (fun b ↦ f.val (representative b)) := by | |
| simp only [smoothScalarTranslate_descent] | |
| exact boundaryZeroGerm_homeomorph (Homeomorph.addRight (quotientMap a)) S | |
| (fun b ↦ f.val (representative b)) | |
| theorem isMinimizerAt_Jacobi_zero_germ_closed {V : ℝ} (hV : V∈Ioo 0 1) | |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) | |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) | |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) | |
| (hp : ∀ m x,N (x+latticeVector m)=N x) | |
| (haeN : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) | |
| (hjN : ∀ w : Cover,∀ v∈surfaceH1 μ,surfaceJacobiB μ (surfaceCurvatureLp μ N hN) | |
| (surfaceSmoothJet μ (directionalNormalScalar N hN hp w)) v=0) | |
| (f : smoothPeriodicScalars) | |
| (hjf : ∀ v∈surfaceH1 μ,surfaceJacobiB μ (surfaceCurvatureLp μ N hN) | |
| (surfaceSmoothJet μ f) v=0) | |
| (x : Torus) (hx : x∈frontier (openRepresentative E)) | |
| (hcl : x∈closure (frontier (openRepresentative E)∩ | |
| boundaryZeroGerm (frontier (openRepresentative E)) (fun x ↦ f.val (representative x)))) : | |
| x∈boundaryZeroGerm (frontier (openRepresentative E)) (fun x ↦ f.val (representative x)) := by | |
| let a : Cover := representative x-cubeCenter | |
| have hshift : quotientMap cubeCenter+quotientMap a=x := by | |
| dsimp [a] | |
| rw [quotientMap_sub,quotientMap_representative] | |
| abel | |
| let E' : Set Torus := (fun b ↦ b+quotientMap a) ⁻¹' E | |
| let μ' := μ.map (translateNormal a) | |
| let N' := normalTranslateField a N | |
| let hN' := normalTranslateField_smooth a hN | |
| let hp' := normalTranslateField_periodic a hp | |
| let f' := sibling_smoothScalarTranslate a f | |
| have hE' : MeasurableSet E' := hE.preimage (measurable_id.add measurable_const) | |
| have hmin' : IsMinimizerAt V E' := isMinimizerAt_translation hmin (quotientMap a) | |
| have hμ' : IsNormalMeasure E' μ' := isNormalMeasure_translation hE hμ a | |
| have haeN' := normalTranslateField_ae μ a hN.continuous hp haeN | |
| have hS' : frontier (openRepresentative E')=(fun b ↦ b+quotientMap a) ⁻¹' | |
| frontier (openRepresentative E) := frontier_openRepresentative_translation hE _ | |
| have hcenter : quotientMap cubeCenter∈frontier (openRepresentative E') := by | |
| rw [hS'] | |
| change quotientMap cubeCenter+quotientMap a∈frontier (openRepresentative E) | |
| rwa [hshift] | |
| obtain ⟨c⟩ := isMinimizerAt_centeredJacobiChart hV hE' hmin' hcenter hμ' hN' hp' haeN' | |
| have hcl' : quotientMap cubeCenter∈closure (frontier (openRepresentative E')∩ | |
| boundaryZeroGerm (frontier (openRepresentative E')) (fun b ↦ f'.val (representative b))) := by | |
| rw [hS',boundaryZeroGerm_translation,←preimage_inter] | |
| change quotientMap cubeCenter∈closure ((Homeomorph.addRight (quotientMap a)) ⁻¹' _) | |
| rw [←(Homeomorph.addRight (quotientMap a)).preimage_closure] | |
| change quotientMap cubeCenter+quotientMap a∈closure _ | |
| rwa [hshift] | |
| have hz := c.zero_germ_closed hE' hμ' hN' hp' haeN' (surfaceCurvatureLp μ' N' hN') f' | |
| (normalTranslate_smooth_weakJacobi μ a hN hp f hjf) (by | |
| rw [normalTranslate_directional a hN hp] | |
| exact normalTranslate_smooth_weakJacobi μ a hN hp _ (hjN c.n)) hcl' | |
| rw [hS',boundaryZeroGerm_translation] at hz | |
| change quotientMap cubeCenter+quotientMap a∈boundaryZeroGerm _ _ at hz | |
| rwa [hshift] at hz | |
| theorem isMinimizerAt_Jacobi_identity {V : ℝ} (hV : V∈Ioo 0 1) | |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) | |
| (hconn : IsPreconnected (frontier (openRepresentative E))) | |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) | |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) | |
| (hp : ∀ m x,N (x+latticeVector m)=N x) | |
| (haeN : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) | |
| (hjN : ∀ w : Cover,∀ v∈surfaceH1 μ,surfaceJacobiB μ (surfaceCurvatureLp μ N hN) | |
| (surfaceSmoothJet μ (directionalNormalScalar N hN hp w)) v=0) | |
| (f : smoothPeriodicScalars) | |
| (hjf : ∀ v∈surfaceH1 μ,surfaceJacobiB μ (surfaceCurvatureLp μ N hN) | |
| (surfaceSmoothJet μ f) v=0) | |
| (hpatch : (frontier (openRepresentative E)∩ | |
| boundaryZeroGerm (frontier (openRepresentative E)) (fun x ↦ f.val (representative x))).Nonempty) : | |
| ∀ x∈frontier (openRepresentative E),f.val (representative x)=0 := by | |
| have hsub := preconnected_subset_of_relative_germ_closed hconn (boundaryZeroGerm_open _ _) hpatch | |
| (fun x hx ↦ isMinimizerAt_Jacobi_zero_germ_closed hV hE hmin hμ hN hp haeN hjN f hjf x hx.2 hx.1) | |
| exact fun x hx ↦ (hsub hx).self_of_nhds hx | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| theorem CenteredJacobiChart.zero_germ_of_local_nonneg | |
| {E : Set Torus} (hE : MeasurableSet E) | |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) | |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) | |
| (hp : ∀ m x,N (x+latticeVector m)=N x) | |
| (haeN : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) | |
| (c : CenteredJacobiChart E μ N) | |
| (q : Lp ℝ ∞ (μ : Measure NormalBundle)) (f : smoothPeriodicScalars) | |
| (hjf : ∀ v∈surfaceH1 μ,surfaceJacobiB μ q (surfaceSmoothJet μ f) v=0) | |
| (hjw : ∀ v∈surfaceH1 μ,surfaceJacobiB μ q | |
| (surfaceSmoothJet μ (directionalNormalScalar N hN hp c.n)) v=0) | |
| (hpos : ∀ᶠ x in 𝓝 (quotientMap cubeCenter),x∈frontier (openRepresentative E) → | |
| 0≤f.val (representative x)) | |
| (hz : f.val cubeCenter=0) : | |
| quotientMap cubeCenter∈boundaryZeroGerm (frontier (openRepresentative E)) | |
| (fun x ↦ f.val (representative x)) := by | |
| let F : Plane → ℝ := fun y ↦ c.weight y*f.val (c.parameter y) | |
| have hF : ContDiff ℝ (⊤ : ℕ∞) F := c.weight_smooth.mul (f.property.1.comp c.parameter_smooth) | |
| have hpos' : ∀ᶠ y in 𝓝 (0 : Plane),0≤f.val (c.parameter y) := by | |
| have ht : Tendsto (fun y ↦ quotientMap (c.parameter y)) (𝓝 (0 : Plane)) | |
| (𝓝 (quotientMap cubeCenter)) := by | |
| have ht := (show Continuous (fun y : Plane ↦ quotientMap (c.parameter y)) from | |
| continuous_quotientMap.comp c.parameter_smooth.continuous).tendsto (0 : Plane) | |
| rw [c.parameter_zero] at ht | |
| exact ht | |
| filter_upwards [ht.eventually hpos,Metric.ball_mem_nhds (0 : Plane) c.r_pos] with y hy hr | |
| simpa only [periodic_representative f.property.2] using hy (c.boundary_disk y (Metric.ball_subset_closedBall hr)) | |
| obtain ⟨s,hs,hsp⟩ := Metric.eventually_nhds_iff.mp hpos' | |
| let r := min s (c.r/2) | |
| have hr : 0<r := lt_min hs (half_pos c.r_pos) | |
| have hrr : r≤c.r := (min_le_right _ _).trans (by linarith [c.r_pos]) | |
| have hL := c.classical_ground_state hE hμ hN hp haeN q f hjf hjw | |
| have hzero := cmc_strong_minimum (plane_gradient_contDiff c.smooth) hF | |
| Metric.isOpen_ball (convex_ball (0 : Plane) r).isPreconnected | |
| (fun y hy ↦ (c.slope_disk y (Metric.ball_subset_closedBall | |
| (Metric.ball_subset_ball hrr hy))).trans (by norm_num : (1/4:ℝ)≤1)) | |
| (fun y hy ↦ mul_nonneg (cmcWeight_pos _).le (hsp (lt_of_lt_of_le hy (min_le_left _ _)))) | |
| (fun y hy ↦ hL y (Metric.ball_subset_ball (min_le_right _ _) hy)) | |
| (Metric.mem_ball_self hr) (show F 0=0 by simp [F,c.parameter_zero,hz]) | |
| apply c.zero_trace_germ f | |
| filter_upwards [Metric.ball_mem_nhds (0 : Plane) hr] with y hy | |
| exact (mul_eq_zero.mp (hzero y hy)).resolve_left (cmcWeight_pos _).ne' | |
| theorem isMinimizerAt_Jacobi_strong_minimum {V : ℝ} (hV : V∈Ioo 0 1) | |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) | |
| (hconn : IsPreconnected (frontier (openRepresentative E))) | |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) | |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) | |
| (hp : ∀ m x,N (x+latticeVector m)=N x) | |
| (haeN : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) | |
| (hjN : ∀ w : Cover,∀ v∈surfaceH1 μ,surfaceJacobiB μ (surfaceCurvatureLp μ N hN) | |
| (surfaceSmoothJet μ (directionalNormalScalar N hN hp w)) v=0) | |
| (f : smoothPeriodicScalars) | |
| (hjf : ∀ v∈surfaceH1 μ,surfaceJacobiB μ (surfaceCurvatureLp μ N hN) | |
| (surfaceSmoothJet μ f) v=0) | |
| {x : Torus} (hx : x∈frontier (openRepresentative E)) | |
| (hpos : ∀ᶠ y in 𝓝 x,y∈frontier (openRepresentative E) → 0≤f.val (representative y)) | |
| (hz : f.val (representative x)=0) : | |
| ∀ y∈frontier (openRepresentative E),f.val (representative y)=0 := by | |
| let a : Cover := representative x-cubeCenter | |
| have hshift : quotientMap cubeCenter+quotientMap a=x := by | |
| dsimp [a] | |
| rw [quotientMap_sub,quotientMap_representative] | |
| abel | |
| let E' : Set Torus := (fun b ↦ b+quotientMap a) ⁻¹' E | |
| let μ' := μ.map (translateNormal a) | |
| let N' := normalTranslateField a N | |
| let hN' := normalTranslateField_smooth a hN | |
| let hp' := normalTranslateField_periodic a hp | |
| let f' := sibling_smoothScalarTranslate a f | |
| have hE' : MeasurableSet E' := hE.preimage (measurable_id.add measurable_const) | |
| have hmin' : IsMinimizerAt V E' := isMinimizerAt_translation hmin (quotientMap a) | |
| have hμ' : IsNormalMeasure E' μ' := isNormalMeasure_translation hE hμ a | |
| have haeN' := normalTranslateField_ae μ a hN.continuous hp haeN | |
| have hS' : frontier (openRepresentative E')=(fun b ↦ b+quotientMap a) ⁻¹' | |
| frontier (openRepresentative E) := frontier_openRepresentative_translation hE _ | |
| have hcenter : quotientMap cubeCenter∈frontier (openRepresentative E') := by | |
| rw [hS'] | |
| change quotientMap cubeCenter+quotientMap a∈frontier (openRepresentative E) | |
| rwa [hshift] | |
| obtain ⟨c⟩ := isMinimizerAt_centeredJacobiChart hV hE' hmin' hcenter hμ' hN' hp' haeN' | |
| have hpos' : ∀ᶠ b in 𝓝 (quotientMap cubeCenter),b∈frontier (openRepresentative E') → | |
| 0≤f'.val (representative b) := by | |
| have ht : Tendsto (fun b : Torus ↦ b+quotientMap a) (𝓝 (quotientMap cubeCenter)) (𝓝 x) := by | |
| simpa only [hshift] using | |
| (show Continuous (fun b : Torus ↦ b+quotientMap a) from continuous_id.add continuous_const).continuousAt.tendsto (x := quotientMap cubeCenter) | |
| filter_upwards [ht.eventually hpos] with b hb hB | |
| rw [hS'] at hB | |
| simpa only [f',smoothScalarTranslate_descent] using hb hB | |
| have hz' : f'.val cubeCenter=0 := by | |
| have he := smoothScalarTranslate_descent a f (quotientMap cubeCenter) | |
| rw [periodic_representative f'.property.2,hshift] at he | |
| exact he.trans hz | |
| have hg := c.zero_germ_of_local_nonneg hE' hμ' hN' hp' haeN' (surfaceCurvatureLp μ' N' hN') f' | |
| (normalTranslate_smooth_weakJacobi μ a hN hp f hjf) (by | |
| rw [normalTranslate_directional a hN hp] | |
| exact normalTranslate_smooth_weakJacobi μ a hN hp _ (hjN c.n)) hpos' hz' | |
| rw [hS',boundaryZeroGerm_translation] at hg | |
| change quotientMap cubeCenter+quotientMap a∈boundaryZeroGerm _ _ at hg | |
| rw [hshift] at hg | |
| exact isMinimizerAt_Jacobi_identity hV hE hmin hconn hμ hN hp haeN hjN f hjf ⟨x,hx,hg⟩ | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| theorem isMinimizerAt_center_normal_unique {V : ℝ} (hV : V∈Ioo 0 1) | |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) | |
| (hcenter : quotientMap cubeCenter∈frontier (openRepresentative E)) : | |
| ∃ w : Cover,∀ (μ : FiniteMeasure NormalBundle),IsNormalMeasure E μ → | |
| ∀ (N : Cover → Cover),ContDiff ℝ (⊤ : ℕ∞) N → | |
| (∀ m x,N (x+latticeVector m)=N x) → | |
| (∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) → N cubeCenter=w := by | |
| obtain ⟨n,g,r,H,hn,hr,hg,hg0,hgc,hi,hcover,_,_⟩ := | |
| isMinimizerAt_smooth_CMC hV hE hmin (quotientMap cubeCenter) hcenter | |
| obtain ⟨f,hf,hfg⟩ := smooth_graph_extension hr hgc | |
| have hcover' (x : Cover) (hx : x∈Metric.ball (0 : Cover) (r/2)) : | |
| quotientMap (cubeCenter+x)∈frontier (openRepresentative E) ↔ | |
| normalGraph n g (normalPlaneCoordinates n x)=x := by | |
| rw [quotientMap_add] | |
| exact hcover x ((Metric.ball_subset_ball (by linarith)) hx) | |
| obtain ⟨R,ρ,U,hRr,_,_,_,hR,hρ,hgC,hU,hρU,hUR,hUO,_,hphase⟩ := | |
| actual_centered_graph_phase_patch hV hE hmin hcenter hn (by linarith : 0<r/2) | |
| hg hg0 ((hgc.of_le (by simp)).mono (Metric.ball_subset_ball (by linarith))) hcover' | |
| let φ := radialHeightCutoff cubeCenter ρ | |
| have hφ : ContDiff ℝ (⊤ : ℕ∞) φ := radialDensityWeight_contDiff _ _ _ | |
| have hsφ : HasCompactSupport φ := radialDensityWeight_hasCompactSupport (by positivity) (by positivity) | |
| have hφs : tsupport φ⊆Metric.closedBall cubeCenter (2*ρ) := | |
| radialDensityWeight_tsupport (by positivity) (by positivity) | |
| have hφU := hφs.trans hρU | |
| have hφO := hφU.trans hUO | |
| have hφR := hφU.trans (hUR.trans Metric.ball_subset_closedBall) | |
| have hφb (x : Cover) : φ x∈Icc 0 1 := | |
| ⟨radialDensityWeight_nonneg _ _ _ _,radialDensityWeight_le_one _ _ _ _⟩ | |
| have hdc (y : Plane) (hy : y∈Metric.closedBall (0 : Plane) R) : ContinuousAt (fderiv ℝ g) y := | |
| (hgC y hy).continuousAt_fderiv (by norm_num) | |
| have hNfg (x : Cover) (hx : x∈tsupport φ) : | |
| graphUnitNormalExtension cubeCenter n f x=graphUnitNormalExtension cubeCenter n g x := by | |
| have hpp : normalPlaneCoordinates n (x-cubeCenter)∈Metric.ball (0 : Plane) (r/2) := by | |
| rw [Metric.mem_ball,dist_zero_right] | |
| have hnorm : ‖x-cubeCenter‖<R := by | |
| simpa only [Metric.mem_ball,dist_eq_norm] using hUR (hφU hx) | |
| exact (normalPlaneCoordinates_norm_le _ _).trans_lt (hnorm.trans hRr) | |
| have he : f=ᶠ[𝓝 (normalPlaneCoordinates n (x-cubeCenter))]g := by | |
| filter_upwards [Metric.isOpen_ball.mem_nhds hpp] with y hy | |
| exact hfg hy | |
| simp only [graphUnitNormalExtension,gradient,he.fderiv_eq] | |
| have hsign : ∃ σ : ℝ,(σ=1 ∨ σ= -1) ∧ | |
| ∀ (μ : FiniteMeasure NormalBundle),IsNormalMeasure E μ → | |
| ∀ᵐ z ∂(μ : Measure NormalBundle),0<φ z.1 → | |
| σ • graphUnitNormalExtension cubeCenter n f z.1=(z.2 : Cover) := by | |
| rcases hphase with hphase|hphase | |
| · have hh : ∀ᵐ x : Cover ∂volume,x∈U → | |
| (quotientMap x∈E ↔ ⟪n,x-cubeCenter⟫<graphHeightExtension cubeCenter n g x) := by | |
| simpa only [graphSignedHeight,sub_neg] using hphase | |
| refine ⟨1,Or.inl rfl,?_⟩ | |
| intro μ hμ | |
| filter_upwards [normalMeasure_graph_normal_ae hE hμ cubeCenter hn hg hdc hh | |
| hφ hsφ hφU hφO hφR hφb] with z hz hpos | |
| rw [one_smul,hNfg _ (subset_tsupport _ hpos.ne')] | |
| exact (hz hpos).symm | |
| · have hh : ∀ᵐ x : Cover ∂volume,x∈U → | |
| (x∈quotientMap ⁻¹' E ↔ graphHeightExtension cubeCenter n g x<⟪n,x-cubeCenter⟫) := by | |
| simpa only [graphSignedHeight,sub_pos,mem_preimage] using hphase | |
| have hh' := graph_superlevel_compl_ae cubeCenter hn hg.continuous hh | |
| refine ⟨-1,Or.inr rfl,?_⟩ | |
| intro μ hμ | |
| have hh'' := normalMeasure_graph_normal_ae hE.compl (isNormalMeasure_compl hE hμ) | |
| cubeCenter hn hg hdc hh' hφ hsφ hφU hφO hφR hφb | |
| have hm : ∀ᵐ z ∂(μ : Measure NormalBundle),0<φ z.1 → | |
| -(z.2 : Cover)=graphUnitNormalExtension cubeCenter n g z.1 := | |
| ae_of_ae_map continuous_reverseNormal.measurable.aemeasurable hh'' | |
| filter_upwards [hm] with z hz hpos | |
| rw [neg_one_smul,hNfg _ (subset_tsupport _ hpos.ne'),← hz hpos,neg_neg] | |
| obtain ⟨σ,hσ,hσae⟩ := hsign | |
| refine ⟨σ • graphUnitNormalExtension cubeCenter n f cubeCenter,?_⟩ | |
| intro μ hμ N hN hp hae | |
| let F : Cover → Cover := fun x ↦ φ x • (N x-σ • graphUnitNormalExtension cubeCenter n f x) | |
| have hF : ContDiff ℝ (⊤ : ℕ∞) F := hφ.smul | |
| (hN.sub ((graphUnitNormalExtension_contDiff _ hn hf).const_smul σ)) | |
| have hsF : HasCompactSupport F := hsφ.smul_right | |
| have hFO : tsupport F⊆{x | ∀ i,0<x i ∧ x i<1} := | |
| (tsupport_smul_subset_left _ _).trans hφO | |
| have he := latticePeriodize_eventuallyEq hsF hFO | |
| (show ∀ i,0≤cubeCenter i ∧ cubeCenter i≤1 by intro i; norm_num [cubeCenter]) | |
| have hz : ∀ᵐ z ∂(μ : Measure NormalBundle),latticePeriodize F z.1=(0 : Cover) := by | |
| filter_upwards [hae,hσae μ hμ] with z hz hloc | |
| rw [(latticePeriodize_eventuallyEq hsF hFO (compactCell_coordinates z.1)).eq_of_nhds] | |
| change φ z.1 • (N z.1-σ • graphUnitNormalExtension cubeCenter n f z.1)=0 | |
| by_cases hpos : 0<φ z.1 | |
| · rw [hz,hloc hpos,sub_self,smul_zero] | |
| · have hzφ : φ z.1=0 := le_antisymm (le_of_not_gt hpos) (hφb z.1).1 | |
| rw [hzφ,zero_smul] | |
| have hb (x : Cover) (hx : quotientMap x∈frontier (openRepresentative E)) : | |
| latticePeriodize F x=0 := | |
| isMinimizerAt_periodic_ae_eq_boundary hV hE hmin hμ | |
| (latticePeriodize_contDiff hF hsF).continuous continuous_const | |
| (latticePeriodize_periodic F) (fun _ _ ↦ rfl) hz x hx | |
| have hzero := hb cubeCenter hcenter | |
| rw [he.self_of_nhds] at hzero | |
| change φ cubeCenter • (N cubeCenter-σ • graphUnitNormalExtension cubeCenter n f cubeCenter)=0 at hzero | |
| have hone : φ cubeCenter=1 := radialHeightCutoff_one hρ (by simp [hρ.le]) | |
| rw [hone,one_smul] at hzero | |
| exact sub_eq_zero.mp hzero | |
| theorem isMinimizerAt_normal_unique {V : ℝ} (hV : V∈Ioo 0 1) | |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) | |
| (x : Cover) (hx : quotientMap x∈frontier (openRepresentative E)) : | |
| ∃ w : Cover,∀ (μ : FiniteMeasure NormalBundle),IsNormalMeasure E μ → | |
| ∀ (N : Cover → Cover),ContDiff ℝ (⊤ : ℕ∞) N → | |
| (∀ m x,N (x+latticeVector m)=N x) → | |
| (∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) → N x=w := by | |
| let a := x-cubeCenter | |
| let E' := (fun z : Torus ↦ z+quotientMap a) ⁻¹' E | |
| have hE' : MeasurableSet E' := (continuous_add_const _).measurable hE | |
| have hmin' : IsMinimizerAt V E' := isMinimizerAt_translation hmin _ | |
| have hc : quotientMap cubeCenter∈frontier (openRepresentative E') := by | |
| rw [frontier_openRepresentative_translation hE] | |
| change quotientMap cubeCenter+quotientMap a∈frontier (openRepresentative E) | |
| rw [←quotientMap_add,show cubeCenter+a=x by dsimp [a]; abel] | |
| exact hx | |
| obtain ⟨w,hw⟩ := isMinimizerAt_center_normal_unique hV hE' hmin' hc | |
| refine ⟨w,?_⟩ | |
| intro μ hμ N hN hp hae | |
| let N' : Cover → Cover := fun y ↦ N (y+a) | |
| have hN' : ContDiff ℝ (⊤ : ℕ∞) N' := hN.comp (contDiff_id.add contDiff_const) | |
| have hp' : ∀ m y,N' (y+latticeVector m)=N' y := by | |
| intro m y | |
| change N ((y+latticeVector m)+a)=N (y+a) | |
| rw [add_right_comm,hp] | |
| have hae' : ∀ᵐ z ∂((μ.map (translateNormal a)) : Measure NormalBundle),N' z.1=(z.2 : Cover) := by | |
| change ∀ᵐ z ∂Measure.map (translateNormal a) (μ : Measure NormalBundle),N' z.1=(z.2 : Cover) | |
| apply (ae_map_iff (measurable_translateNormal a).aemeasurable | |
| (isClosed_eq (hN'.continuous.comp normalEmbedding_continuous.fst) continuous_normalUnit).measurableSet).mpr | |
| filter_upwards [hae] with z hz | |
| change N' (translateNormal a z).1=(z.2 : Cover) | |
| rw [translateNormal_periodic_eval hp'] | |
| change N (((z.1 : Cover)-a)+a)=(z.2 : Cover) | |
| simpa only [sub_add_cancel] using hz | |
| have hh := hw (μ.map (translateNormal a)) (isNormalMeasure_translation hE hμ a) | |
| N' hN' hp' hae' | |
| change N (cubeCenter+a)=w at hh | |
| simpa only [show cubeCenter+a=x by dsimp [a]; abel] using hh | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| def sibling_reflectNormal (z : NormalBundle) : NormalBundle := | |
| (cubeRepresentative (torusFirstReflection (quotientMap z.1)), | |
| ⟨firstReflection (z.2 : Cover),by rw [LinearIsometryEquiv.norm_map,z.2.property]⟩) | |
| lemma sibling_measurable_reflectNormal : Measurable sibling_reflectNormal := by | |
| apply Measurable.prodMk | |
| · exact measurable_cubeRepresentative.comp (torusFirstReflection.continuous.measurable.comp | |
| (measurable_quotientMap.comp (measurable_subtype_coe.comp measurable_fst))) | |
| · exact (firstReflection.continuous.measurable.comp | |
| (measurable_subtype_coe.comp measurable_snd)).subtype_mk | |
| lemma sibling_reflectNormal_periodic_eval {Y : Type*} {F : Cover → Y} | |
| (hp : ∀ m x,F (x+latticeVector m)=F x) (z : NormalBundle) : | |
| F (sibling_reflectNormal z).1=F (firstReflection z.1) := by | |
| change F (representative (torusFirstReflection (quotientMap z.1)))=_ | |
| rw [←firstReflection_quotient,periodic_representative hp] | |
| theorem isNormalMeasure_firstReflection {E : Set Torus} (hE : MeasurableSet E) | |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) : | |
| IsNormalMeasure (torusFirstReflection ⁻¹' E) (μ.map sibling_reflectNormal) := by | |
| constructor | |
| · have hm : ((μ.map sibling_reflectNormal).mass : ℝ)=(μ.mass : ℝ) := by | |
| have hh := congrArg ENNReal.toReal (Measure.map_apply sibling_measurable_reflectNormal | |
| MeasurableSet.univ (μ := (μ : Measure NormalBundle))) | |
| simpa only [preimage_univ,FiniteMeasure.mass,FiniteMeasure.coeFn_def, | |
| FiniteMeasure.toMeasure_map,Measure.real,ENNReal.toReal] using hh | |
| rw [hm,hμ.1,perimeter_firstReflection hE] | |
| · intro F hF | |
| change (∫ z,normalTest F hF.1.continuous z | |
| ∂Measure.map sibling_reflectNormal (μ : Measure NormalBundle))=_ | |
| rw [integral_map sibling_measurable_reflectNormal.aemeasurable | |
| (normalTest F hF.1.continuous).continuous.aestronglyMeasurable, | |
| flux_firstReflection hF hE,←hμ.2 _ (reflectionPullback_admissible hF)] | |
| apply integral_congr_ae | |
| exact Eventually.of_forall fun z ↦ by | |
| change ⟪firstReflection (z.2 : Cover),F (sibling_reflectNormal z).1⟫= | |
| ⟪(z.2 : Cover),firstReflection (F (firstReflection z.1))⟫ | |
| rw [sibling_reflectNormal_periodic_eval hF.2.1] | |
| simpa only [firstReflection_involutive] using | |
| (firstReflection.inner_map_map (z.2 : Cover) (firstReflection (F (firstReflection z.1)))) | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| lemma graph_tangent_of_graph_containment {n m : Cover} (hm : ‖m‖=1) | |
| {f g : Plane → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f) | |
| (hg : ContDiff ℝ (⊤ : ℕ∞) g) (hf0 : f 0=0) | |
| (he : ∀ᶠ y in 𝓝 (0 : Plane), | |
| normalGraph m g (normalPlaneCoordinates m (normalGraph n f y))=normalGraph n f y) | |
| (w : Plane) : ⟪graphSlopeNormal m (gradient g 0),fderiv ℝ (normalGraph n f) 0 w⟫=0 := by | |
| have hF := normalGraph_contDiff hf (n := n) | |
| have hG := normalGraph_contDiff hg (n := m) | |
| have hF0 : normalGraph n f 0=0 := by | |
| change (normalPlaneLinearIsometry n) 0+f 0 • n=0 | |
| rw [map_zero,hf0,zero_smul,add_zero] | |
| have hdP := (normalPlaneCoordinates m).hasFDerivAt.comp 0 (hF.differentiable (by simp) 0).hasFDerivAt | |
| have hdG := ((hG.differentiable (by simp) _).hasFDerivAt.comp 0 hdP).fderiv | |
| have he' : (fun y ↦ normalGraph m g (normalPlaneCoordinates m (normalGraph n f y)))=ᶠ[𝓝 (0 : Plane)]normalGraph n f := he | |
| have hd := he'.fderiv_eq (𝕜 := ℝ) | |
| change fderiv ℝ (normalGraph m g ∘ normalPlaneCoordinates m ∘ normalGraph n f) 0=_ at hd | |
| rw [hdG] at hd | |
| change (fderiv ℝ (normalGraph m g) (normalPlaneCoordinates m (normalGraph n f 0))).comp | |
| ((normalPlaneCoordinates m).comp (fderiv ℝ (normalGraph n f) 0))=fderiv ℝ (normalGraph n f) 0 at hd | |
| rw [hF0,map_zero] at hd | |
| rw [←hd,ContinuousLinearMap.comp_apply] | |
| exact normalGraph_tangent_normal hm hg 0 _ | |
| lemma graphSlopeNormal_gradient_zero {n : Cover} (hn : ‖n‖=1) : graphSlopeNormal n (0 : Plane)=n := by | |
| rw [graphSlopeNormal_weight hn] | |
| change (cmcWeight 0)⁻¹ • (n-(normalPlaneLinearIsometry n) 0)=n | |
| simp [cmcWeight] | |
| theorem normalized_CMC_common_germ_normals {B C : Set Torus} | |
| {N M : Cover → Cover} {H K : ℝ} | |
| (hBch : HasNormalizedCMCCharts B N H) (hCch : HasNormalizedCMCCharts C M K) | |
| {a : Cover} (ha : quotientMap a∈B) | |
| (he : ∀ᶠ z in 𝓝 (quotientMap a),z∈B ↔ z∈C) : N a=M a ∨ N a= -M a := by | |
| obtain ⟨hn,f,hf,hf0,hdf0,hfm,_⟩ := hBch a ha | |
| obtain ⟨hm,g,hg,hg0,hdg0,hgm,hgc⟩ := hCch a (he.self_of_nhds.mp ha) | |
| have hF0 : normalGraph (N a) f 0=0 := by | |
| change (normalPlaneLinearIsometry (N a)) 0+f 0 • N a=0 | |
| rw [map_zero,hf0,zero_smul,add_zero] | |
| have hFt : Tendsto (normalGraph (N a) f) (𝓝 (0 : Plane)) (𝓝 (0 : Cover)) := by | |
| simpa only [hF0] using ((normalGraph_contDiff hf (n := N a)).continuous.continuousAt (x := 0)).tendsto | |
| have hq : Tendsto (fun y : Plane ↦ quotientMap (a+normalGraph (N a) f y)) | |
| (𝓝 0) (𝓝 (quotientMap a)) := by | |
| have hc : Continuous (fun y : Plane ↦ quotientMap (a+normalGraph (N a) f y)) := | |
| continuous_quotientMap.comp (continuous_const.add (normalGraph_contDiff hf).continuous) | |
| simpa only [hF0,add_zero] using (hc.continuousAt (x := 0)).tendsto | |
| have hcontain : ∀ᶠ y in 𝓝 (0 : Plane), | |
| normalGraph (M a) g (normalPlaneCoordinates (M a) (normalGraph (N a) f y))=normalGraph (N a) f y := by | |
| filter_upwards [hfm,hq.eventually he,hFt.eventually hgc] with y hfy hey hgy | |
| exact hgy.mp (hey.mp hfy.2.1) | |
| have ht (w : Plane) : ⟪M a,fderiv ℝ (normalGraph (N a) f) 0 w⟫=0 := by | |
| simpa only [hdg0,graphSlopeNormal_gradient_zero hm] using | |
| graph_tangent_of_graph_containment hm hf hg hf0 hcontain w | |
| have hcol := graph_normal_tangent_vector_eq hn hf 0 ht | |
| rw [hdf0,graphSlopeNormal_gradient_zero hn] at hcol | |
| have hnorm := congrArg norm hcol | |
| rw [hm,norm_smul,Real.norm_eq_abs,hn,mul_one] at hnorm | |
| rcases (abs_eq (by norm_num : (0 : ℝ)≤1)).mp hnorm.symm with hplus|hminus | |
| · rw [hplus,one_smul] at hcol | |
| exact Or.inl hcol.symm | |
| · rw [hminus,neg_one_smul] at hcol | |
| exact Or.inr (by rw [hcol,neg_neg]) | |
| theorem normalized_CMC_common_germ_oriented {B C : Set Torus} | |
| {N M : Cover → Cover} (hN : Continuous N) (hM : Continuous M) | |
| (hpN : ∀ m x,N (x+latticeVector m)=N x) | |
| (hpM : ∀ m x,M (x+latticeVector m)=M x) | |
| {H K : ℝ} (hBch : HasNormalizedCMCCharts B N H) (hCch : HasNormalizedCMCCharts C M K) | |
| {x : Torus} (hx : x∈B) (he : ∀ᶠ z in 𝓝 x,z∈B ↔ z∈C) | |
| (hNM : N (representative x)=M (representative x)) : | |
| x∈orientedSurfaceAgreement B C N M := by | |
| have hn : ‖N (representative x)‖=1 := (hBch _ (by simpa only [quotientMap_representative] using hx)).1 | |
| have hc : Continuous (fun z : Torus ↦ ⟪N (representative z),M (representative z)⟫) := | |
| (continuous_periodic_descent hN hpN).inner (continuous_periodic_descent hM hpM) | |
| have hpos : ∀ᶠ z in 𝓝 x,0<⟪N (representative z),M (representative z)⟫ := by | |
| apply hc.continuousAt.eventually (lt_mem_nhds ?_) | |
| change 0<⟪N (representative x),M (representative x)⟫ | |
| rw [←hNM,real_inner_self_eq_norm_sq,hn,one_pow]; norm_num | |
| filter_upwards [he,he.eventually_nhds,hpos] with z hze hzeg hzpos | |
| refine ⟨hze,fun hzB ↦ ?_⟩ | |
| have hzB' : quotientMap (representative z)∈B := by simpa only [quotientMap_representative] using hzB | |
| rcases normalized_CMC_common_germ_normals hBch hCch hzB' | |
| (by simpa only [quotientMap_representative] using hzeg) with hsame|hop | |
| · exact hsame | |
| · have hm := (hCch (representative z) (by simpa only [quotientMap_representative] using hze.mp hzB)).1 | |
| rw [hop,inner_neg_left,real_inner_self_eq_norm_sq,hm,one_pow] at hzpos | |
| linarith | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| theorem HasNormalizedCMCCharts.neg {B : Set Torus} {N : Cover → Cover} {H : ℝ} | |
| (hch : HasNormalizedCMCCharts B N H) (hN : ContDiff ℝ (⊤ : ℕ∞) N) | |
| (hbd : ∀ a : Cover,quotientMap a∈B → | |
| (∀ᶠ v in 𝓝 a,‖N v‖=1) ∧ divergence N a=H) : | |
| HasNormalizedCMCCharts B (fun x ↦ -N x) (-H) := by | |
| intro a ha | |
| obtain ⟨hn,f,hf,hf0,hdf0,hfm,hfc⟩ := hch a ha | |
| have hm : ‖-N a‖=1 := by simpa only [norm_neg] using hn | |
| have ht : ⟪graphSlopeNormal (N a) (gradient f 0),-N a⟫≠0 := by | |
| rw [hdf0,graphSlopeNormal_gradient_zero hn,inner_neg_right,real_inner_self_eq_norm_sq,hn,one_pow] | |
| norm_num | |
| obtain ⟨g,I,hg,hg0,hIc,hI0,hgf,hcov⟩ := sibling_normalGraph_regraph hn hm hf hf0 ht | |
| have hnmatch : ∀ᶠ y in 𝓝 (0 : Plane),-N (a+normalGraph (N a) f y)= | |
| (-1 : ℝ) • graphSlopeNormal (N a) (gradient f y) := by | |
| filter_upwards [hfm] with y hy | |
| rw [hy.1,neg_one_smul] | |
| have hnorm := sibling_normalGraph_regraph_normal hn hm hf hg hf0 hg0 hIc hI0 hgf hN.continuous.neg | |
| rfl (Or.inr rfl) hnmatch | |
| have hi : Tendsto I (𝓝 (0 : Plane)) (𝓝 (0 : Plane)) := by simpa only [hI0] using hIc.tendsto | |
| have hmem : ∀ᶠ y in 𝓝 (0 : Plane),quotientMap (a+normalGraph (-N a) g y)∈B := by | |
| filter_upwards [hgf,hi.eventually hfm] with y hy hmy | |
| rw [hy]; exact hmy.2.1 | |
| refine ⟨hm,g,hg,hg0,hnorm.1,?_,?_⟩ | |
| · filter_upwards [hnorm.2,hmem,hnorm.2.eventually_nhds] with y hy hby hny | |
| refine ⟨hy,hby,?_⟩ | |
| have hny' : ∀ᶠ z in 𝓝 y,-N (a+normalGraph (-N a) g z)= | |
| (1 : ℝ) • graphSlopeNormal (-N a) (gradient g z) := by simpa only [one_smul,Pi.neg_apply] using hny | |
| have hd := graph_normal_extension_divergence a hm hg y (Or.inl rfl) hN.neg | |
| (by simpa only [norm_neg] using (hbd _ hby).1) hny' | |
| rw [divergence_neg,(hbd _ hby).2] at hd | |
| change -H= -(1 : ℝ)*planeCMC g y at hd | |
| linarith | |
| · filter_upwards [hfc,hcov] with v hv hc | |
| exact hv.trans hc.symm | |
| theorem CMC_surfaces_unoriented_identity {B C : Set Torus} (hBc : IsClosed B) (hCc : IsClosed C) | |
| (hBconn : IsPreconnected B) (hCconn : IsPreconnected C) | |
| {N M : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) (hM : ContDiff ℝ (⊤ : ℕ∞) M) | |
| (hpN : ∀ m x,N (x+latticeVector m)=N x) | |
| (hpM : ∀ m x,M (x+latticeVector m)=M x) | |
| {H K : ℝ} (hBch : HasNormalizedCMCCharts B N H) (hCch : HasNormalizedCMCCharts C M K) | |
| (hMbd : ∀ a : Cover,quotientMap a∈C → | |
| (∀ᶠ v in 𝓝 a,‖M v‖=1) ∧ divergence M a=K) | |
| (hpatch : ∃ x∈B,∀ᶠ z in 𝓝 x,z∈B ↔ z∈C) : | |
| B=C ∧ ((H=K ∧ ∀ x∈B,N (representative x)=M (representative x)) ∨ | |
| (H= -K ∧ ∀ x∈B,N (representative x)= -M (representative x))) := by | |
| obtain ⟨x,hx,hxe⟩ := hpatch | |
| have hx' : quotientMap (representative x)∈B := by simpa only [quotientMap_representative] using hx | |
| have hxg : ∀ᶠ z in 𝓝 (quotientMap (representative x)),z∈B ↔ z∈C := by | |
| simpa only [quotientMap_representative] using hxe | |
| rcases normalized_CMC_common_germ_normals hBch hCch hx' hxg with hsame|hop | |
| · have hag := normalized_CMC_common_germ_oriented hN.continuous hM.continuous hpN hpM hBch hCch hx hxe hsame | |
| obtain ⟨he,hHK,hNM⟩ := CMC_surfaces_identity hBc hCc hBconn hCconn hN.continuous hM.continuous | |
| hpN hpM hBch hCch ⟨x,hx,hag⟩ | |
| exact ⟨he,Or.inl ⟨hHK,hNM⟩⟩ | |
| · have hneg := hCch.neg hM hMbd | |
| have hnegp : ∀ m x,-M (x+latticeVector m)= -M x := fun m x ↦ congrArg Neg.neg (hpM m x) | |
| have hag := normalized_CMC_common_germ_oriented hN.continuous hM.continuous.neg hpN hnegp hBch hneg hx hxe hop | |
| obtain ⟨he,hHK,hNM⟩ := CMC_surfaces_identity hBc hCc hBconn hCconn hN.continuous hM.continuous.neg | |
| hpN hnegp hBch hneg ⟨x,hx,hag⟩ | |
| exact ⟨he,Or.inr ⟨hHK,hNM⟩⟩ | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| theorem CMC_surface_oriented_subset {B C : Set Torus} (hCc : IsClosed C) | |
| (hBconn : IsPreconnected B) | |
| {N M : Cover → Cover} (hN : Continuous N) (hM : Continuous M) | |
| (hpN : ∀ m x,N (x+latticeVector m)=N x) | |
| (hpM : ∀ m x,M (x+latticeVector m)=M x) | |
| {H K : ℝ} (hBch : HasNormalizedCMCCharts B N H) (hCch : HasNormalizedCMCCharts C M K) | |
| (hpatch : (B∩orientedSurfaceAgreement B C N M).Nonempty) : | |
| B⊆orientedSurfaceAgreement B C N M ∧ H=K := by | |
| obtain ⟨x,hx,hag⟩ := hpatch | |
| have he := normalized_CMC_common_patch_curvature hpN hpM hBch hCch hx hag | |
| subst K | |
| exact ⟨preconnected_subset_of_relative_germ_closed hBconn (isOpen_orientedSurfaceAgreement _ _ _ _) | |
| ⟨x,hx,hag⟩ (fun _ hy ↦ orientedSurfaceAgreement_closed_at hCc hN hM hpN hpM hBch hCch hy.2 hy.1),rfl⟩ | |
| theorem CMC_surface_unoriented_subset {B C : Set Torus} (hCc : IsClosed C) | |
| (hBconn : IsPreconnected B) | |
| {N M : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) (hM : ContDiff ℝ (⊤ : ℕ∞) M) | |
| (hpN : ∀ m x,N (x+latticeVector m)=N x) | |
| (hpM : ∀ m x,M (x+latticeVector m)=M x) | |
| {H K : ℝ} (hBch : HasNormalizedCMCCharts B N H) (hCch : HasNormalizedCMCCharts C M K) | |
| (hMbd : ∀ a : Cover,quotientMap a∈C → | |
| (∀ᶠ v in 𝓝 a,‖M v‖=1) ∧ divergence M a=K) | |
| (hpatch : ∃ x∈B,∀ᶠ z in 𝓝 x,z∈B ↔ z∈C) : | |
| ∀ x∈B,∀ᶠ z in 𝓝 x,z∈B ↔ z∈C := by | |
| obtain ⟨x,hx,hxe⟩ := hpatch | |
| have hx' : quotientMap (representative x)∈B := by simpa only [quotientMap_representative] using hx | |
| have hxg : ∀ᶠ z in 𝓝 (quotientMap (representative x)),z∈B ↔ z∈C := by | |
| simpa only [quotientMap_representative] using hxe | |
| rcases normalized_CMC_common_germ_normals hBch hCch hx' hxg with hsame|hop | |
| · have hag := normalized_CMC_common_germ_oriented hN.continuous hM.continuous hpN hpM hBch hCch hx hxe hsame | |
| have he := CMC_surface_oriented_subset hCc hBconn hN.continuous hM.continuous hpN hpM hBch hCch ⟨x,hx,hag⟩ | |
| exact fun y hy ↦ (he.1 hy).mono (fun _ h ↦ h.1) | |
| · have hneg := hCch.neg hM hMbd | |
| have hnegp : ∀ m x,-M (x+latticeVector m)= -M x := fun m x ↦ congrArg Neg.neg (hpM m x) | |
| have hag := normalized_CMC_common_germ_oriented hN.continuous hM.continuous.neg hpN hnegp hBch hneg hx hxe hop | |
| have he := CMC_surface_oriented_subset hCc hBconn hN.continuous hM.continuous.neg hpN hnegp hBch hneg ⟨x,hx,hag⟩ | |
| exact fun y hy ↦ (he.1 hy).mono (fun _ h ↦ h.1) | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| theorem isMinimizerAt_lifted_boundary_global {V : ℝ} (hV : V∈Ioo 0 1) | |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) (a : Cover) : | |
| a∈coverEssentialBoundary (quotientMap ⁻¹' E) ↔ | |
| quotientMap a∈frontier (openRepresentative E) := by | |
| let b := a-cubeCenter | |
| let F := (fun z : Torus ↦ z+quotientMap b) ⁻¹' E | |
| have hF : MeasurableSet F := hE.preimage (continuous_id.add continuous_const).measurable | |
| have hminF := isMinimizerAt_translation hmin (quotientMap b) | |
| have hcent := isMinimizerAt_lifted_boundary hV hF hminF (x := cubeCenter) (by intro i; norm_num [cubeCenter]) | |
| have he : quotientMap ⁻¹' F=(Homeomorph.addRight b) ⁻¹' (quotientMap ⁻¹' E) := by | |
| ext x | |
| change quotientMap x+quotientMap b∈E ↔ quotientMap (x+b)∈E | |
| rw [quotientMap_add] | |
| rw [he,cover_boundary_measurePreserving_homeomorph (Homeomorph.addRight b) | |
| (measurePreserving_add_right volume b),frontier_openRepresentative_translation hE] at hcent | |
| have hb : cubeCenter+b=a := by dsimp [b]; abel | |
| change cubeCenter+b∈coverEssentialBoundary (quotientMap ⁻¹' E) ↔ | |
| quotientMap cubeCenter+quotientMap b∈frontier (openRepresentative E) at hcent | |
| rw [←quotientMap_add,hb] at hcent | |
| exact hcent | |
| lemma lifted_ae_eq_torus {E F : Set Torus} (hE : MeasurableSet E) (hF : MeasurableSet F) | |
| (h : quotientMap ⁻¹' E=ᵐ[volume]quotientMap ⁻¹' F) : E=ᵐ[torusVolume]F := by | |
| have hr : ∀ᵐ x ∂volume.restrict fundamentalCube,quotientMap x∈E ↔ quotientMap x∈F := by | |
| filter_upwards [ae_restrict_of_ae h] with x hx | |
| exact Iff.of_eq hx | |
| have hm : MeasurableSet {x : Torus | x∈E ↔ x∈F} := by measurability | |
| have ht := (ae_map_iff measurable_quotientMap.aemeasurable hm).mpr hr | |
| change ∀ᵐ x ∂torusVolume,x∈E ↔ x∈F at ht | |
| filter_upwards [ht] with x hx | |
| exact propext hx | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| def coverAEAgreement (A B : Set Cover) : Set Cover := | |
| (volume.restrict {x | ¬(x∈A ↔ x∈B)}).supportᶜ | |
| lemma isOpen_coverAEAgreement (A B : Set Cover) : IsOpen (coverAEAgreement A B) := | |
| Measure.isOpen_compl_support | |
| lemma mem_coverAEAgreement_iff {A B : Set Cover} {x : Cover} : | |
| x∈coverAEAgreement A B ↔ ∃ O : Set Cover,IsOpen O ∧ x∈O ∧ | |
| ∀ᵐ y ∂volume,y∈O → (y∈A ↔ y∈B) := by | |
| change x∉(volume.restrict {y | ¬(y∈A ↔ y∈B)}).support ↔ _ | |
| rw [(nhds_basis_opens x).notMem_measureSupport] | |
| constructor | |
| · rintro ⟨O,⟨hx,hO⟩,hz⟩ | |
| refine ⟨O,hO,hx,?_⟩ | |
| rw [ae_iff] | |
| rw [Measure.restrict_apply hO.measurableSet] at hz | |
| convert hz using 1 | |
| congr 1 | |
| ext y | |
| simp only [mem_ofPred_eq,Classical.not_imp,mem_inter_iff] | |
| · rintro ⟨O,hO,hx,hae⟩ | |
| refine ⟨O,⟨hx,hO⟩,?_⟩ | |
| rw [Measure.restrict_apply hO.measurableSet] | |
| rw [ae_iff] at hae | |
| convert hae using 1 | |
| congr 1 | |
| ext y | |
| simp only [mem_ofPred_eq,Classical.not_imp,mem_inter_iff] | |
| lemma coverAEAgreement_trans {A B C : Set Cover} {x : Cover} | |
| (h : x∈coverAEAgreement A B) (k : x∈coverAEAgreement B C) : | |
| x∈coverAEAgreement A C := by | |
| obtain ⟨O,hO,hx,hae⟩ := mem_coverAEAgreement_iff.mp h | |
| obtain ⟨P,hP,hxp,hbe⟩ := mem_coverAEAgreement_iff.mp k | |
| apply mem_coverAEAgreement_iff.mpr | |
| refine ⟨O∩P,hO.inter hP,⟨hx,hxp⟩,?_⟩ | |
| filter_upwards [hae,hbe] with y hy hz hOP | |
| exact (hy hOP.1).trans (hz hOP.2) | |
| lemma coverAEAgreement_symm {A B : Set Cover} {x : Cover} | |
| (h : x∈coverAEAgreement A B) : x∈coverAEAgreement B A := by | |
| obtain ⟨O,hO,hx,hae⟩ := mem_coverAEAgreement_iff.mp h | |
| exact mem_coverAEAgreement_iff.mpr ⟨O,hO,hx,hae.mono (fun _ h hy ↦ (h hy).symm)⟩ | |
| lemma coverAEAgreement_compl {A B : Set Cover} {x : Cover} | |
| (h : x∈coverAEAgreement A B) : x∈coverAEAgreement Aᶜ Bᶜ := by | |
| obtain ⟨O,hO,hx,hae⟩ := mem_coverAEAgreement_iff.mp h | |
| exact mem_coverAEAgreement_iff.mpr ⟨O,hO,hx,hae.mono (fun _ h hy ↦ not_congr (h hy))⟩ | |
| lemma coverAEAgreement_disjoint {A B : Set Cover} (hA : MeasurableSet A) (hB : MeasurableSet B) : | |
| Disjoint (coverAEAgreement A B) (coverAEAgreement A Bᶜ) := by | |
| have hm : MeasurableSet {x | ¬(x∈A ↔ x∈B)} := by measurability | |
| have he : {x : Cover | ¬(x∈A ↔ x∈Bᶜ)}={x | ¬(x∈A ↔ x∈B)}ᶜ := by | |
| ext x; simp only [mem_ofPred_eq,mem_compl_iff]; tauto | |
| rw [Set.disjoint_iff_inter_eq_empty] | |
| change (volume.restrict {x | ¬(x∈A ↔ x∈B)}).supportᶜ ∩ | |
| (volume.restrict {x | ¬(x∈A ↔ x∈Bᶜ)}).supportᶜ=∅ | |
| rw [he,←compl_union,←Measure.support_add,Measure.restrict_add_restrict_compl hm, | |
| Measure.support_eq_univ,compl_univ] | |
| lemma ae_eq_of_coverAEAgreement_univ {A B : Set Cover} | |
| (h : coverAEAgreement A B=univ) : A=ᵐ[volume]B := by | |
| have hs : (volume.restrict {x | ¬(x∈A ↔ x∈B)}).support=∅ := by | |
| have hh := congrArg (fun S : Set Cover ↦ Sᶜ) h | |
| simpa only [coverAEAgreement,compl_compl,compl_univ] using hh | |
| have hz := Measure.restrict_eq_zero.mp (Measure.support_eq_empty_iff.mp hs) | |
| have hae : ∀ᵐ x ∂volume,x∈A ↔ x∈B := ae_iff.mpr hz | |
| filter_upwards [hae] with x hx | |
| exact propext hx | |
| theorem cover_local_phase_global {A B : Set Cover} (hA : MeasurableSet A) (hB : MeasurableSet B) | |
| (hcov : ∀ x,x∈coverAEAgreement A B ∨ x∈coverAEAgreement A Bᶜ) : | |
| A=ᵐ[volume]B ∨ A=ᵐ[volume]Bᶜ := by | |
| by_cases hn : (coverAEAgreement A B).Nonempty | |
| · left | |
| have hsub : (univ : Set Cover)⊆coverAEAgreement A B := | |
| isPreconnected_univ.subset_left_of_subset_union (isOpen_coverAEAgreement _ _) | |
| (isOpen_coverAEAgreement _ _) (coverAEAgreement_disjoint hA hB) | |
| (fun x _ ↦ hcov x) (by simpa only [univ_inter] using hn) | |
| exact ae_eq_of_coverAEAgreement_univ (univ_subset_iff.mp hsub) | |
| · right | |
| apply ae_eq_of_coverAEAgreement_univ | |
| apply eq_univ_of_forall | |
| intro x | |
| exact (hcov x).resolve_left (fun hx ↦ hn ⟨x,hx⟩) | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| end CubicTorus | |
| end | |
| end | |
| end | |
| end OAI | |